Exercise 1, § 1

Let E be the vector space of functions $x \mapsto g(x) = \int_0^x f(t)\, dt$ defined on the interval $[0,1]$ of $\mathbf{R}$, where $f$ runs over the set of regulated functions on $[0,1]$. Let P be the set of increasing functions belonging to E. Show that P is a convex cone such that $E = P - P$, $P \cap (-P) = \{0\}$, and that E, equipped with the order structure defined by the relation $g - h \in P$, is a Riesz space.